Prior to modern science, this was the case in natural philosophy. Physics was where this changed, the signal event being the publication of Newton’s Principia in 1687. By describing a logical structure leading to an unending supply of precisely verifiable and useful conclusions, it gave the word theory a new meaning. Modern physical theories--Electromagnetic, Quantum, Relativity, etc.--are all based on overwhelming amounts of hard data. They are not speculative at all, and, as modern technology demonstrates, they apply to the real world (in the popular sense of real). This chapter describes how large amounts of data were first reduced to theory.

The data that physical theory is based upon, is called ‘hard’ because it has negligible uncertainty due to unknown or uncontrolled influences. A ball of iron may be very precisely observed to fall faster than one of wood, but the precision of the observation do not suffice to make it useful. The observation must also include information about environmental factors such as air density and air velocity. The air is part of the environment of the system being studied. Its neglect in this cases causes the data to be inaccurate even when precise. Data contingent on the environment of the system being studies are not useful unless those conditions are controlled and their effects understood.

Isolated systems are those that can be accurately studied without paying attention to their environment. Although a system is never completely isolated, it can be effectively so if environmental influences on it are either undetectable, or can be understood as causing measurements to have statistical fluctuations.

Systems found to be effectively isolated in their natural state were the first to be understood scientifically studied and understood. For example, light passing through air over short distances (meters rather then kilometers), is effectively uninfluenced by the surrounding air; and early technology was capable of controlling the curvature and position of opaque, transparent, or reflecting surfaces put in light's path. Consequently, geometrical optics was one of the first branches of true science--true, in that experiment led to quantitative laws of reflection and refraction; experiment, in that observations were for the specific purpose of measuring of angles of reflection and refraction by the controlled placement of the various geometrical surfaces to which these laws refer.

But optical experiment was exceptional; early technology just happened to be capable of handling its experimental requirements; similarly, vibrating strings and mechanical levers. There were few others, and there was no effort made to advance technology for this purpose. The idea of experiment had not taken hold.

We know that quantitative experiments were done by Archimedes (who served as the inspiration for Galileo), and later by medieval natural philosophers, and undoubtedly others lost in history, but these starts were generally disparaged or ignored by colleagues: the better minds obstructed the best.

It was mankind's good fortune to have, standing free and apart from this situation, literally and figuratively, the great system of the heavens, especially the solar system. The latter's interaction with its environment is negligible because it is imbedded in a vacuum and is far removed from other major gravitational systems (the stars). It drew attention to itself because of both its drama and practical importance. It provided large amounts of quantitative data with no experiment. In astronomy, humanity was given its first non-contingent and therefore accurate set of data about a physical system.

Gods were identified with heavenly bodies that moved along their stately course as if in obedience to higher law, as if unperturbed by the sudden unexpected contingencies that everything on Earth is prone to. It was daily observation of the heavens that impressed on inhabitants of the Near East (where the nightly heavenly display is spectacular) that a higher law might rule over Earth.

So closely associated with this kind of motion that we hardly pay any conscious attention to it is the idea of simplicity. Lawful, regular, unperturbed motion is generally equated with simple motion. Simple means on the one hand easy to analyze, having few underlying factors (each mathematically described by a parameter) determining it, which most often results in it having few twists and turns in its space-time behavior. In effect, lawfulness is intuitively associated with that which might be totally analyzed in terms of--reduced to--a few parameters. It is in part because of this intuition that astronomy played its historic role.

Astronomers had to learn to reduce, and thus make sense of, large sets of hard data. The earliest large compilations of astronomical data were Babylonian. They were organized qualitatively in terms of constellations, signs of the zodiac, stations of the moon, and so on:

Qualitative descriptions of astronomical data slowly became quantitative. The oldest example found so far dates from 523 BCE, but it must have been the result of a long prior evolution. It records the delay between the setting of the sun and the rising of the moon, and the setting of the moon and rising of the sun. People eventually saw how to organize the tables so that their regularities came to the fore. This last step produced the earliest form of quantitative physical theory.

Table 1 shows a portion of a Babylonian table of longitudes of Jupiter when it was in a certain position relative to the sun occurring at regular intervals a little over a year apart. Each longitude was measured relative to that of the Zodiac named in the first column. The third column shows how much the Longitude changed between observations; its entries equal 30° + the difference between consecutive Longitudes because the 12 signs of the Zodiac were exactly 30° apart in angle. The Difference column records differences between consecutive entries in the Motion column; its entries are therefore differences of differences (called second differences).

The Difference column alternates between sequences of ± 1° 48’ values except for isolated smaller intermediate values which for simplicity we will ignore in this discussion. For example, looking at the third and second entries in the Motion column, one sees that 30° 26’ - 28° 38’ = +1° 48’, and that is the entry between them in the Difference column. Starting with one entry in the Motion column, and knowing the Differences, all other entries in the Motion column could be reconstructed by simple additions, e.g. 30° 26’ = 28° 38’ +1° 48’. Since the Motion column tells how much motion there has been between successive Longitudes, it can be used to reconstruct the Longitudes in a similar way.

Thus the whole table could be reconstructed, by successive additions, simply from knowing that the last column consisted of alternating sequences of ± 1° 48’ (six in each sequence) and knowing the starting values in the Motion and Longitude columns. These relatively few facts could replace an effectively endless table; the table was reducible to them. Such tables constituted humanity’s first quantitative theories.

We can call the theory itself the statement that second differences of longitudes measured at fixed times of the day and year are constant except for regularly spaced changes in sign. This theory might be applied to many different heavenly bodies. Each application would need a pair of parameters and a pair of initial conditions similar to those of Jupiter which the table shows to be (1° 48’ and 6) and (21° 49’ and 29° 41'), respectively. The initial conditions are the initial entries in the 'longitude' and 'motion' columns of the table. Starting from these initial conditions, the theory prescribes how to build-up a never ending table of future and past observations by adding and subtracting the parameters of the system. An unending list of observations is reduced to a few numbers and equations.

In contrast to this, Greek astronomy created geometric theories. Suppose you made a sequence of observations of position. The first four might look as shown in the Figure 1. Although they have some obvious regularity–the second column goes up as the third column goes down–it would be difficult to guess, just looking at the numbers, what the next row of numbers would be. But after plotting and seeing the first four points, you could probably quickly guess about where the next one or two might be thanks to our superior biologically inbuilt abilities at visual analysis. After about fifteen more observations revealing further points along the circle, most people would confidently bet that they continued, equally spaced, around the whole circle.

This leads to the simplest theory of heavenly motion, that it proceeds at a uniform rate about circles in the sky. Any applications needs the following parameters: the radius of the circle, the position of its center, and the fixed distance and time between consecutive points. An initial condition is specified by where a particular heavenly body is on the circle at some initial time. This again leads to a never ending table of predictions of past and future observations.

As we shall now see, the use of geometry proved to be enormously advantageous. As its observational precision improved, theory had to become more refined and the refinements required constructions that depended on geometrical intuition. With their strictly numerical methods, the Babylonians could never have come up with them.

The solar system is a stellar teacher of mathematical physics, and this is largely due to its hierarchy of simplicities and complexities. Systems without such an hierarchy are very hard to analyze. Physics has unfolded like a sequence of well put problems in a text. Each chapter’s problems, have been able to be solved, but then have also been found to have some subtle, interesting, and apparently minor complications designed to draw the student onward, to make the student think and read further.

One simplicity is the shortness of the day compared to the year. The size of the relevant factor, about 365 (days per year), helps us to easily separate two kinds of phenomena which we might otherwise confuse. We easily distinguish daily from yearly cycles: the daily rotations of the sun and stars from the yearly rotation of seasons. Had the year been only a few long days long (perfectly possible), combined effects of daily and yearly rotations could have been far more complicated and difficult to distinguish.

This is suggested by the graphs in Figure 3 each of which is the sum of three oscillations. When sums of oscillations with well separated periods (separated like the solar system’s daily, monthly or yearly periods) are examined, each contribution to the sum can be made out as in the lower curve. However, if as in the upper figure, this is not the case, the sum behaves wildly and individual contributions to it cannot be easily made out. This would be the case for the solar system if, in analogy, the day, month and year were not very different periods of time.

The visual dominance of only a few major celestial groupings: the sun, the planets, the moon, and the fixed stars provide yet another simplicity. Humanity was not confused by many motions to analyze simultaneously which, for example, could well have been the case were Earth surrounded by many moons.

One of the solar system’s first noticeable complications came from the fact that the length of days, months, and years are not related as simple fractions. This has practical consequences. The year is approximately 365 1/4 days. If this were exact, then in every 4 years there would be exactly 1461 days. By simply adding a day every leap year–the scheme of the Julian calendar–days and years would march exactly in step forever. But this is not the case; the year is shorter than this by about 1 day every 128 years. This correction was added in to form the Gregorian calendar in 1582. It does not solve the problem completely. There is in fact absolutely no way days and years can ever be kept exactly in step by any scheme. People are often surprised that the solar system’s natural periods–day, month, year–are not related by integer ratios (rational). In fact, if these numbers were rational, that would be very surprising and very significant indeed.

Now 1 day in 128 years is a small correction and so it took a while before it was detected. After the Dark Age, about 13 centuries after Julius Caesar inaugurated the Julian Calendar (named for him but the brain work was Greek), Europe awoke from its sleep and discovered an accumulated 8 day discrepancy in the vernal equinox, the start of spring. Along with similar lunar problems this caused great consternation in the Church. Easter was being celebrated on the wrong days (to the fiendish delight of the pagans). The Church was drawn on to study this small complication more carefully and consequently gave critical support to astronomy. Popes and Emperors consulted astronomers. They accumulated measurements unmatched in number and accuracy.

The orbits of heavenly bodies provided the most well known hierarchy of simplicities and complexities. First came the essentially circular orbits of sun, moon and stars in relation to which the complex unexplained planetary orbits of the planets were of minor significance. Then, the planets were simply included as essentially circular orbits by Copernicus, leaving the various minor unexplained deviations from circularity. These were greatly simplified by Kepler's introduction of elliptical orbits. Kepler also introduced the start of the idea of dynamics which brought to fruition by Newton who explained all the orbits simple and nearly perfectly. Finally, a small imperfection in the Newtonian theory, it explanation of orbital precession, required Einstein's gravitational theory.

The explanation of each minor unexplained complexity required major revolutions in physical theory. These, in turn, profoundly influenced thought beyond physics. This illustrates an important lesson. Human perception and cognition has evolved to serve our immediate biological needs. We see well close, less well far. What appears small may be large but distant; what appears minor in human perspective, has often turned out to be major in cosmological perspective. It is only science that can provide the latter. However well science may seem to explain everything, however minor the remaining discrepancies, there is every reason to believe that their minor nature is only an artifact of their distance from the current range of scientific instruments and human needs. Huge changes in the scientific picture of the world will probably always be just below the horizon.

This explains an apparent paradox. On the one hand, the effectiveness of science steadily increases. It is a stable, unidirectional increase, an accumulation of knowledge. Our calculation methods become ever more refined, and able to explain ever more previously unexplainable minor discrepancies in their predictions. On the other hand, in the very face of this stability, the picture of the world science provides is highly unstable, or has seemed so far, as complete conceptual revolutions--Copernican, Newtonian, Einsteinian--have followed one another in the relatively short space of a few centuries.

Claudius Ptolemy, an ethnic Greek working in Alexandria Egypt (circa 150 CE) did his best known work in astronomy, music, geography and astrology; his books became Europe’s basic texts after the Dark Ages. Ptolemaic astronomy, based on a research program proposed by Plato, was described in Mathematikei Syntaxis called by the Arabs, Al-Magesti (the greatest) and then transliterated by the Europeans as Almagest. It served as the basis of astronomy until about a half a century after 1543, the year Copernicus (known as the second Ptolemy) published De Revolutionibus (The Revolutions).

Ptolemy assumed that the Earth’s surface was fixed in space. The sun, stars and planets, on the other hand, were attached to a celestial sphere that rotated daily about the Earth at its center. Because of this, every heavenly body seemed to move at a steady rate around a circular orbit in the sky once per day. But experience showed that this could not be all. If, for example, the sun performed only its daily rotation, then there would be no seasons, and it would rise and set at the same two points on the horizon every day of the year. The sun’s circle moved in a yearly cycle.

The planets moved in a more complicated manner. Figure 4 shows Mars’ apparent position in the sky on the first of the months indicated. The picture is taken at the same time on each of these days. If the only motion of a heavenly body were just a daily circuit it would return to the same point on that circle at the same time each day. This means that any change in the position of the planets and constellations shown in the picture must be due to motion they undergo in addition to their daily circular motion.

Looking at pictures taken exactly one day apart cancels out the daily circular motion of the heavens (that is, the effect of the Earth’s daily hour rotation about its own axis as understood from the Copernican viewpoint). From now on, the discussion will be only about heavenly motion beyond this daily rotation. It will be understood that comparing observations taken at the same time each day has factored the daily rotation out.

Figure 4 shows how Mars moves relative to the constellations. Like all the planets, it occasionally wanders a bit across the constellations. The planets follow an almost regular course, partially but not completely bound by necessity--somewhat like humans except that planets have a more serene, and of course elevated life.

The method used by the ancients to describe the motions of the planets is credited to Plato and is implied by his phrasing of the problem:

Uniform and ordered motion" means circular motion at constant speed. The use of the plural, ‘motions’, signifies a compounding of two or more uniform circular motions. How compounding took place is described by Figure 3.

There are a number of good reasons for conceiving of motion purely in terms of uniform circular motions added to one another. The most important of these is uniform circular motion was, to the ancients, a form of natural motion: it could sustain itself indefinitely without the intervention of an external agent. Once initiated at the start of the universe, it could keep going forever. Plato's intuition was one of wheels on wheels--a small wheel (epicycle) on a large one, both going forever, with the trajectory of the smaller one going in an orbit which was therefore not circular.

No one had ever seen a wheel here on Earth moving for very long times in uniform circular motion, but they had seen motion suggestive of this and were able to imagine the rest. The designation 'natural' contained the seeds of the idea that would eventually lead to death of animism. Wheels slowed down--they died--because everything here on Earth is susceptible to 'corruption', to decay and death (today we use the words friction and entropy). Thus, in the heavens, freed of the corruption of their earthly surrounding, they might well go on forever.

That something needed life to move ("move" means to change in any way) seemed obvious, but uniform circular motion did not seem to need a conscious intelligence. The reason is that such motion was completely predictable and that was so because it was so simple. To say something is in uniform circular motion, with a speed ( e.g. of one orbit per day, or per month, or per year), with a position of its center ( e.g. the center of the Earth), and with a radius (not exactly known but known to exist), is to reduce all the observations of the circle to these few parameters and words. And by virtue of this simple reduction, it is to be able to predict that thing's future. If you can predict it with certainty, it does not (seem to) need intelligence.

Thus, as long as planetary motion could not be predicted, planets seemed somewhat like wandering Olympian gods. On the other hand the stars did not wander and thus had little resemblance to gods aside from being immortal. Early on, they were conceived simply as holes in a rotating sphere letting through light from an eternal fire.

Figure 5 does not describe the complete Ptolemaic model of the skies but it is a first step. It approximates centuries of observational data by a few parameters: the radii and rotational rates of deferents and epicycles. To improve agreement with data, Ptolemy would have needed additional geometrical constructions to combine with epicycles. With each construction there would be new parameters. Thus, to make a more accurate theory, he would have to use more parameters. The values of these parameters for each heavenly body were set independently. Parameters of different planets were unrelated to one another. They seemed to be connected by no law.

Ptolemaic theory eventually reduced thousands of data to about fifty parameters. It graphically demonstrated how complex planetary motion was explicable in terms of simple motions. In doing so, it permitted the accurate prediction of heavenly motion, a task of immense significance for agriculture, navigation, and religion.

Could this be more generally true? How much of the complexity about us could similarly be reduced to the compounding of simple factors? This is the program of all analysis--the reduction of complexity to simplicity. Here we seen how the starts and stops of the planets, their lifelike motions, were reduced, and in the process, at least some of the apparent life was taken from them; it had an effect similar to opening up a child’s mechanical toy and seeing how simple circular motions of its inner gears combined to imitate life. Thus, the possibilities inherent in Plato’s program had immense intellectual significance. In all these respects, practical and intellectual, Ptolemy’s implementation of Plato’s program was a major achievement and it dominated astronomy for two thousand years.

But it was also a failure. It no longer hoped or attempted to reveal the level of simplicity Plato had dreamed of. It was too ugly to have emanated from the One/Good. It resigned itself to being only a very human tool to predict future astronomical data. This changed goal was called ‘saving the phenomena’.

The hypotheses referred to are those for which Copernicus is now famous: his assertion that the Earth rotates about its axis, and that it and the planets revolve about the sun. Although the quote only says that the hypotheses need not be true, it was clearly addressed to those people who were quite sure they were not true. The writer wanted to assure them that the book was nonetheless useful because the hypotheses were merely a calculational device. He was assuring them that Copernicanism just saved phenomena; it was just phenomenology.

If the Earth is rotating, commonsense would indicate that it would cause a mighty wind and it would spin things off.

The Earth’s daily rotation about its axis is illustrated in 6 where the sky has been reduced to a single star which, if one supposes Earth to be motionless, appears to rotate daily in a circle. An observer on Earth would see the same thing if the Earth were rotating about its axis in the opposite sense. Either hypothesis explains the pure observation–the relative motion of observer and Earth and star–equally well. In a calculational sense, there is a sort of simplification in assuming that it is only the Earth. But it made no physical sense. If the Earth rotated, why did things not fly off it? Why was there no wind as it spun through air? Why did buildings not sway and fall as when Earth moved in a quake? Why not believe the obvious testimony of our senses that place Earth firmly immobile beneath our feet? No one, including Copernicus, could answer such objections. And on top of all this, a moving Earth contradicted the literal interpretation of the Bible then being particularly insisted upon by the Church for reasons that will be discussed in the chapter on the trial of Galileo.

Copernicus and Ptolemy differed not only concerning Earth’s daily rotation but also concerning geocentrism (Figure 5) versus heliocentrism (Figure 7). Either system would have been able to be made to agree with the data with approximately equal accuracy. This is symbolized by the identical planetary trajectory shown on the right of either figure. The systems differed in the mathematical constructions (described in the figure) used to produce this agreement.

Copernicus became famous as the ‘second Ptolemy’ within a few years of his death. This was due, first of all, to the fact that his work was the first systematic attempt to fit the data in centuries and therefore could take into account the data accumulated in that period. Doing the analysis using any system, heliocentric or geocentric, was a major accomplishment. Copernicus’ also became famous because he eliminated certain geometrical constructs used in the Ptolemaic system called equants. At the end of his life Copernicus is reported to have said that this elimination, not heliocentrism itself, was his greatest achievement and virtually all of his colleagues agreed. We can hear this in the following statement of Tycho Brahe (1546-1601), the greatest astronomer in the years between Copernicus and Kepler, who, in the fall of 1574 at the University of Copenhagen, started

The unsuitable and offensive hypotheses were the equants. The Alphonsine tables (King Alphonso of Portugal had needed them especially for navigation) were the previous tables which had been based on the Ptolemaic system and which were no longer in sufficient agreement with data.

Tycho, however, goes on as follows:

The ‘irregular’ manner he refers to is the non-uniformity of motion created by equants;

Why did Tycho and Copernicus both believe it absurd to think that planets moved in circles at a non-uniform rate? After all, any of us can turn a wheel at however an irregular rate we might wish. The belief arose out of their fundamental assumption as to the meaning of natural law already discussed but worth a reiteration. Natural law described natural motion and that meant unforced motion, self-sustaining motion not directed by any ‘will, whether that will be of man, gods, or God. The absence of will is the very essence of the definition of Nature.

But there was only one kind of unforced motion known which had this property and that was circular motion: If you set a sufficiently balanced and lubricated wheel in motion it can move uniformly by itself for long periods of time. And it was obvious that deviated from uniformity in motion only to the extent that it was influenced by forces external to it–forces set up by imperfections in its manufacture (imbalanced wheels, unlubricated bearing,…). The planets did not move in uniform circular motion, and that is just why they had always been thought of as conscious beings such as gods. But Plato’s program, which is the program of all science as derived from that of the first natural philosophers, aimed at explaining, naturally, as much motion as possible. Plato aimed to drive the mythology of the gods from the astronomy of the planets–just the converse of thinking of them as the ‘brothers’ of Man.

Nature was that portion of phenomena–motion–that could be explained naturally. The more explained naturally, the greater the domain of Nature. Thus the complex, lifelike, apparently conscious motion of the planets was to be explained as a clever combination of underlying natural motions–no more godlike than the motion of the toy driven by hidden circular motions. Copernicus had found a way to describe heavenly motion purely in terms of underlying natural —uniform circular–motion. He thereby retained the naturalness of heavenly motion, rescuing it from the hands of gods. In this context, in the battle to create the concept of nature itself, his achievement in eliminating the need for equants truly was important.

This whole worldview was not merely sustained by millennia of tradition, but also by careful observation and thought. That is why the greatest thinkers of the time–just as smart as the greatest thinkers of today–followed it. It was as certain to them as similarly founded beliefs commonly held today are to us. The importance attributed to the elimination of the equant is generally treated as an incomprehensible oddity, a quaint and backward looking adulation of Aristotelian metaphysics. But although Copernicus himself was not so sure of the lasting importance of his heliocentrism (which in any case had already been proposed by Aristarchus, a pupil of Plato), he was sure of the lasting importance of the elimination of equants. Such is the gulf separating our worldview from theirs.

Copernicus, by his hypothesis of Earth’s daily rotation, had connected Astronomy, the first collection of accurate quantitative data, with common sense. Are we living on a whirling merry-go-round!? His hypothesis contradicted our common interpretation of sense data. Most people, including astronomers, would not easily ‘take leave of their sense’ and give up the idea of a motionless Earth.

The quote from Tycho’s lecture shows that the challenge to common sense implicit in the Copernican construction led Tycho to make clear that Mathematical Astronomy was not to be taken seriously as physics. Mathematical astronomy did not describe what ‘really’ happened. Copernicus’ theory was merely a calculational device useful for ‘saving phenomena’; it was merely, as we would say today, phenomenology.

In addition, Copernicus, by his very success, had demonstrated the mathematical equivalence of looking at astronomical phenomena using two very different sets of hypotheses. If there were two, there were probably many. What claim had any of them to be uniquely true? Thus by using physically unreasonable hypotheses and showing how effectively they explained the data, Copernicus seemed to demonstrate the physical unreality of his as well as even future systems of mathematical Astronomy.

We have here an interesting and not so obvious example of a set of decisions all based on the pragmatics of data reduction. Copernican theory was too useful to simply discard. It reduced the newer, enlarged set of astronomical data, and did so in a somewhat cleaner way. But if it were interpreted as implying physical truth, it would have resulted in a net loss of data reduction in the sense that our understanding of all our everyday phenomena–a huge amount of sense data–would have been thrown into disarray. The solution was simply to keep Copernicus, but to consider it as a purely mathematical construction. The drive towards data reduction demanded, at that time, the ambivalent retention of both heliocentric and the geocentric views.

This type of resolution, is not uncommon. Everyone knows of innumerable situations in which the same set of facts can be explained with different hypotheses. As the previous discussion of the meaning of the word theory indicated, that is the usual situation. Historical facts are continually being reinterpreted differently by different historical theories, different story lines connecting the same historical dots. The astronomical facts being discussed here are no different. In fact, it is a modern fashion amongst historians of science to connect the bare facts of the Copernican revolution very differently than they have been here. That is at least one reason for hearing how a physicist would like to connect them.

But this was not the whole story. The parameters Copernicus discovered using his system displayed for the first time what is now called systematics. In his own words in the Ninth Chapter of Book I of De Revolutionibus , Copernicus says:

Here he is pointing out that the planetary years increase with their distance from the sun and that the fixed stars rotate not at all about the sun (thus, as if they are effectively infinitely far away!) This was the first regularity discovered in the heavens since ancient times. It connected different planets’ previously unconnected parameters thereby revealing a coherence extending over the entire solar system. It was the type of connection Plato must have been hoping for–the work of not a blundering but a very subtle Demiurge. And it is the type of connection physicists look for today when they try out different phenomenological analyses of data.

It was a wonderful surprise and inspiration to those few, such as Kepler, who could intuit its deeper significance. They realized that this regularity had a negligible probability of occurring by chance. There must be some deep truth in heliocentrism: something more than mere phenomenology. It must be more than merely a clever way to keep the books on planetary motion. The Platonic idea of a simple mathematical reality behind what we see on the walls of our cave was given a glimmering hope of new life.

[see Figure ] asked Tycho in 1589 in a letter to another astronomer who was espousing Copernican theory. And as for Earth moving around the sun, he calculated on the basis of his own improved observations, that it might imply that some stars could have radii larger than the Earth’s orbit. This seemed clearly ludicrous. So Tycho hired the 29 year old theoretician Kepler to analyze his data in terms of his own more sensible theory. Kepler began work on the orbit of Mars early in 1600.

Kepler had gained the notice of Tycho (and Galileo) with his first published work four years previously, the Mysterium Cosmographicum. With the vision of the great scientist, he had appreciated the significance of Copernicus’ revelation that planetary years increased with distance from the sun. Kepler sensed that such a connection expressed an underlying principle. A theory that quantitatively explained this connection would have probably been generalizable to other parameters, for why would God stop with the radii?

Kepler had more than vision (there are many visionaries), he had the faith, tenacity, rebelliousness and raw talent to pursue his vision. His faith was essentially religious: Pythagoreanism transcribed into a Christian context. It was needed to carry him through many years of arduous labor.

When Kepler attacked the data bequeathed him by Tycho, he developed two ideas central to modern science. Neither was original in conception, both could be traced back millennia, and both were increasingly in the air. But Kepler combined and executed these ideas; he was the first to turn them into demonstrably useful tools for extracting new truths from nature.

The first idea was the application of dynamics to celestial motion: the overthrow of geometry by dynamics. The ancient view of natural law was geometric in that a law was expressed as a prescribed path in space. From the modern dynamical view, objects move in response to a force. Kepler, more than anyone else, changed this fundamental assumption; he hypothesized that something (which we now think of as a force) emanating from the sun caused planetary motion. Force as cause. The difference is crucial because the geometric view suggested looking for simple orbits, whereas the dynamical view caused one to look for simple forces, and neither implies the other.

There is more than one origin to this complex idea. First it is the extension to the heavens of common experience on earth. Common experience tells us that forces on objects–pushes and pulls–cause motion but do not uniquely determine paths. But of course such pushes and pulls as we experience directly by our sense are ours–they are determined by will and are not natural. So it was crucial that force was set upon the road to being a natural entity–set upon the road only because even Kepler himself did not think at first that this force was natural in this respect. He conceived of it as a living creation of the Christian God.

As the study of magnetism had already showed, forces could extend over long distances. Kepler believed (incorrectly) that magnetism was the force acting over the long distance between the sun and its planets. Underlying both ideas was universality: the conviction that one form of natural law applied to both the sub and superlunary world.

Kepler converted this variety of notions into a mathematical requirement he imposed on his analysis of the data: measure all planetary trajectories only from the physical center of the sun–strict heliocentrism. By doing this, he converted a belief into a working hypothesis. He rather than Copernicus actually stuck to this idea as a fixed guide. Lacking the concept of dynamics, Copernicus had fallen back on the geometry of circles as his basic guide and had used the center of Earth’s orbit, not the sun, as the center of the solar system. This turned out to be a crucial technical ingredient in Kepler’s success.

Kepler’s second idea is epitomized by the now famous concluding passage of his Astronomia Nova’s nineteenth chapter.

God and goodness! We pass this over today as a piece of standardized piety, But in the light of the Pagan (which Kepler himself fought most directly) it means a great deal. It means that this small complication of eight minutes should be paid attention to. It is not a discrepancy one might expect from a world that was basically chaotic. It is part of the thoroughgoing lawfulness of the world–its complete order down to the smallest detail. It is part of God’s complete control over existence. It is gift from God to man. It is a slight discrepancy in the correct theory leading us on to the next stage of greater insight--a signpost.

This (religious) reverence for data was, as Kepler says, well repaid. It was also observationally well founded. The 8’ of an arc referred to was a discrepancy between his initial theory and Tycho’s observed orbit of Mars. This angle of 8/60° was smaller than Ptolemy’s precision, which did not exceed 10’. Therefore, when constructing their theories, Copernicus and Ptolemy had both wisely ignored discrepancies of this (and even much greater) size since they could have well been spurious. But 8/60° could be measured by Tycho.

Kepler was the first theoretician with access to such data and had the wit and courage to exploit it by discarding his initial work and starting again. The 8’ thereby became a classic example of a small complication leading to great consequences.

The relevance of dynamics and the reverence for data, though conceptually separable, had to operate together to be useful. Without a faith in the relevance of highly precise data--a faith in a rule of law extending down to the smallest detail, as opposed to a belief in law superimposed upon a foundation of chaos--he would not have grasped at Tycho’s careful observations. Without data accurate enough to distinguish different dynamical effects, Kepler had nothing to guide him beyond uniform circular motion. Without an intuition of a dynamics linked to the sun, he would have had nothing to guide his new method of analyzing the data.. This interplay between faith, intuition, and aesthetics on the one hand, and data on the other, mediated by mathematics, underlies all of modern science.